Question

given the following information, determine which lines, if any, are parallel. state the postulate or theorem that justifies your answer.

1. ∠3≅∠7
2. ∠9≅∠11
3. ∠2≅∠16
4. m∠5 + m∠12 = 180

find x so that ℓ || m. show your work.

1. (2x + 6)° 130°
2. (4x - 10)° (3x + 10)°
3. (4x)° (x + 6)°
4. (7x - 5)° (5x + 19)°
5. (3x + 10)° (5x + 18)°

given the following information, determine which lines, if any, are parallel. state the postulate or theorem that justifies your answer.

1. m∠bcg + m∠fgc = 180
2. ∠cbf≅∠gfh
3. ∠efb≅∠fbc
4. ∠acd≅∠kbf

find x so that l || m. identify the postulate or theorem you used.

1. (4x - 6)° (3x + 6)°
2. (7x - 24)° (5x + 18)° (2x + 12)° (5x - 15)°

Explanation:

Step1: Recall parallel - line postulates and theorems

When two lines are cut by a transversal, certain angle - relationships indicate parallel lines. Corresponding angles, alternate interior angles, alternate exterior angles, and same - side interior angles have specific properties for parallel lines.

Step2: Analyze "Determine which lines, if any, are parallel" problems

For $\angle3\cong\angle7$

These are alternate interior angles. By the Alternate Interior Angles Theorem, if alternate interior angles are congruent, then the lines are parallel. So, $a\parallel b$.

For $\angle9\cong\angle11$

These are corresponding angles. By the Corresponding Angles Postulate, if corresponding angles are congruent, then the lines are parallel. So, $\ell\parallel m$.

For $\angle2\cong\angle16$

These are corresponding angles. By the Corresponding Angles Postulate, $\ell\parallel m$.

For $m\angle5 + m\angle12=180$

These are same - side interior angles. By the Same - Side Interior Angles Theorem, if same - side interior angles are supplementary, then the lines are parallel. So, $a\parallel b$.

Step3: Analyze "Find $x$ so that $\ell\parallel m$" problems

For the first "Find $x$" problem with $(2x + 6)^{\circ}$ and $130^{\circ}$

These are same - side interior angles. For $\ell\parallel m$, they must be supplementary. So, $(2x + 6)+130 = 180$.
$2x+136 = 180$.
$2x=180 - 136$.
$2x = 44$.
$x = 22$.

For the second "Find $x$" problem with $(4x-10)^{\circ}$ and $(3x + 10)^{\circ}$

These are corresponding angles. For $\ell\parallel m$, they must be congruent. So, $4x-10=3x + 10$.
$4x-3x=10 + 10$.
$x = 20$.

For the third "Find $x$" problem with $(6x + 4)^{\circ}$ and $(8x-8)^{\circ}$

These are corresponding angles. For $\ell\parallel m$, $6x + 4=8x-8$.
$8 + 4=8x-6x$.
$2x=12$.
$x = 6$.

For the fourth "Find $x$" problem with $(4x)^{\circ}$ and $(x + 6)^{\circ}$

These are corresponding angles. For $\ell\parallel m$, $4x=x + 6$.
$4x-x=6$.
$3x=6$.
$x = 2$.

For the fifth "Find $x$" problem with $(7x-5)^{\circ}$ and $(5x + 19)^{\circ}$

These are corresponding angles. For $\ell\parallel m$, $7x-5=5x + 19$.
$7x-5x=19 + 5$.
$2x=24$.
$x = 12$.

For the sixth "Find $x$" problem with $(3x + 10)^{\circ}$ and $(5x + 18)^{\circ}$

These are corresponding angles. For $\ell\parallel m$, $3x + 10=5x + 18$.
$10-18=5x-3x$.
$-8 = 2x$.
$x=-4$.

Answer:

1. $a\parallel b$ (Alternate Interior Angles Theorem)
2. $\ell\parallel m$ (Corresponding Angles Postulate)
3. $\ell\parallel m$ (Corresponding Angles Postulate)
4. $a\parallel b$ (Same - Side Interior Angles Theorem)
5. $x = 22$ (Same - Side Interior Angles Theorem)
6. $x = 20$ (Corresponding Angles Postulate)
7. $x = 6$ (Corresponding Angles Postulate)
8. $x = 2$ (Corresponding Angles Postulate)
9. $x = 12$ (Corresponding Angles Postulate)
10. $x=-4$ (Corresponding Angles Postulate)